The case arose after Twitter erupted following the daubing of a memorial to the women of the Second World War in Whitehall with the words "F--- Tory scum" during an anti-austerity demonstration.

Monroe took legal action over what her lawyer told the judge was a "widely published allegation" that she had "either vandalised a war memorial or approved of such an act", an allegation that would "inevitably cause serious damage to reputation".

Jonathan Price, for Hopkins, told the judge her case was "this relatively trivial dispute arose and was resolved on Twitter in a period of several hours".

He argued "no lasting harm, and certainly no serious harm", to Monroe's reputation resulted from it.

He said Hopkins had "mistakenly" used Monroe's Twitter handle instead of that of another columnist who had tweeted about the war memorial incident.

But Mr Justice Warby ruled "whilst the claimant may not have proved that her reputation suffered gravely, I am satisfied that she has established that the publications complained of caused serious harm to her reputation".

He said their publication "not only caused Ms Monroe real and substantial distress, but also harm to her reputation which was serious".

The judge concluded: "Ms Monroe is entitled to fair and reasonable compensation, which I assess at £24,000."

Speaking after the decision was announced, Monroe said: "I am very relieved that it is over and done with.

"It has been a very long and very arduous process. There have been many times when I have almost given up and walked away. But I started something and I had to see it through, and I have done."

Monroe's lawyer, Mark Lewis, a partner at Seddons solicitors, said she had "finally been vindicated in full from the libellous and wholly false accusation by Katie Hopkins that she had supported the vandalisation of a war memorial".

He added: "Jack Monroe never did, and coming from a proud military family, never would. Despite pointing this out to Katie Hopkins within minutes of her first tweet wrongly accusing Jack, Hopkins did not apologise.

"Rather the self-styled 'rent-a-gob' defiantly posted another defamatory tweet. The price of not saying sorry has been very high.

"Despite repeated opportunities to back down, Hopkins obstinately refused to apologise, instead conducting her defence by slinging as much mud as she could to hide the fact that she had made this false allegation."

He added: "The price of not saying sorry has been very high. Hopkins has had to pay out of her own pocket a six-figure sum in damages and costs for a tweet that should have been deleted within minutes as soon as she was told it was wrong. On this occasion, the cost of renting that gob was particularly high.

"Hopkins claimed that Twitter was just the wild west where anything goes. The judge has shown that there is no such thing as a Twitter outlaw."